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The Bernstein homomorphism via Aguiar-Bergeron-Sottile universality
[Submitted on 11 Apr 2016 (v1), last revised 16 Jun 2026 (this v · 2026-06-17 · via math.CO updates on arXiv.org

Mathematics > Combinatorics

arXiv:1604.02969 (math)

[Submitted on 11 Apr 2016 (v1), last revised 16 Jun 2026 (this version, v3)]

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Abstract:If H is a commutative connected graded Hopf algebra over a commutative ring k, then a certain canonical k-algebra homomorphism H -> H (x) QSym is defined, where QSym denotes the Hopf algebra of quasisymmetric functions over k. This homomorphism generalizes the "internal comultiplication" on QSym, and extends what Hazewinkel (in Section 18.24 of his "Witt vectors") calls the Bernstein homomorphism.
We construct this homomorphism with the help of the universal property of QSym as a combinatorial Hopf algebra (a well-known result by Aguiar, Bergeron and Sottile) and extension of scalars (the commutativity of H allows us to consider, for example, H (x) QSym as an H-Hopf algebra, and this change of viewpoint significantly extends the reach of the universal property).
Comments: 46 pages; comments are welcome! Version 3 corrects errors found by GPT-5.5, particularly in the proof of Proposition 3.10
Subjects: Combinatorics (math.CO); Rings and Algebras (math.RA)
MSC classes: 05E05, 16T05
Cite as: arXiv:1604.02969 [math.CO]
  (or arXiv:1604.02969v3 [math.CO] for this version)
  https://doi.org/10.48550/arXiv.1604.02969

arXiv-issued DOI via DataCite

Submission history

From: Darij Grinberg [view email]
[v1] Mon, 11 Apr 2016 14:13:29 UTC (34 KB)
[v2] Tue, 19 Apr 2016 04:39:42 UTC (35 KB)
[v3] Tue, 16 Jun 2026 06:10:09 UTC (37 KB)

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